In practice, \(X^n\)  can take any arbitrary distribution. In this section, we show a method to extend our results for the 1-order homogeneous Markov chain to any general sequence of random variable \(\lbrace X_n\rbrace _{n=1}^{\infty }\) . In the next section, we apply our general bounds for some special cases of \(m\) -order Markov chains or mixed datasets where each feature vector is taken from a Markov chain in a pool of \(|\mathcal {P}|\)  Markov chains. Our development in this section is based on spectral method in [1]}.
